Why Can Positive Average Returns Coexist with Negative Compound Growth? The Arithmetic-Geometric Mean Gap in China's A-Share Market

A positive arithmetic mean return does not imply positive growth of wealth over multiple investment periods. This study employs monthly individual-stock returns of 4,647 A-shares listed on the Shenzhen and Shanghai Stock Exchanges in China from 2015 to 2024, and analyzes the difference between the stock-level arithmetic and geometric mean returns. The arithmetic mean return has a median of 0.78%, while the geometric mean return has a median of -0.13%. For 1,696 stocks, the arithmetic mean return is positive but the geometric mean return is negative (36.50% of the main sample; 45.79% of stocks with positive arithmetic mean returns). There is a significant and positive, but non-causal, relationship between return variance and the arithmetic-geometric mean gap. In a pooled winsorisation sensitivity check at the 1st and 99th percentiles, the variance coefficient rises from 0.147 to 0.462, consistent with the 0.5 coefficient in the small-return second-order approximation. This study demonstrates that average return should be accompanied by compound growth, volatility, and tail risk in long-horizon investment evaluation. But the evidence is not conclusive as to whether the Kelly allocation is optimal for all investors.